Bayes’ Theorem: The Geometry of Changing Beliefs

01Probability Is a Bookkeeping System

Probability is not a force hiding inside an object; it is a quantitative language for uncertainty. A prior records what was plausible before new evidence. A likelihood asks how compatible the evidence is with a candidate explanation. A posterior is the revised distribution after the evidence has been accounted for. The numbers must add to one because the candidates partition the possibilities.

02The Formula in Plain Clothes

Bayes’ theorem is P(A|B) = P(B|A)P(A) / P(B). Read it as a recipe: start with the prior probability of A, multiply by how likely B would be if A were true, then divide by the overall probability of seeing B. The denominator is not decorative. It is the normalization step that prevents one vivid explanation from absorbing more probability than exists.

03The Coin Urn as a Working Model

The documented urn example makes the update tangible. There are two fair coins, two coins that land heads with probability 0.6, and one coin that lands heads with probability 0.9. Before a flip, a randomly selected coin has probabilities 0.4, 0.4, and 0.2 of belonging to those groups. Seeing heads raises the relative standing of the 0.9 group, but not to certainty: the other groups can also produce heads.

04The Base-Rate Trap

A positive test result is evidence, not a diagnosis. If a condition is rare, false positives from the much larger healthy population can outnumber true positives even when a test is highly sensitive and specific. Bayes forces the base rate into the calculation. Ignoring it is the classic error: confusing P(positive|disease) with P(disease|positive).

05The Geometry of Evidence

The 3Blue1Brown video treats belief as a geometric object that can be reshaped by evidence. That intuition is valuable because the update is multiplicative: evidence stretches some hypotheses more than others, then the total is rescaled. In odds form, posterior odds equal prior odds multiplied by a likelihood ratio. Independent evidence can therefore accumulate cleanly—provided the independence assumption is real.

06Where Models Go Wrong

A Bayesian calculation can be exact and still answer the wrong question. A prior may be poorly calibrated; a likelihood model may omit a confounder; two observations may not be independent. The cure is not to abandon Bayes, but to expose assumptions, run sensitivity checks, and report how the posterior changes when reasonable inputs change.

07A Better Habit of Mind

Bayes’ theorem is useful far beyond medicine or coin flips. It is the logic of debugging, forecasting, search, and scientific inference: state what you believed, identify what you observed, and show the update. The discipline is modest but powerful. Confidence should move when evidence moves—and the size of the move should be visible.